Properties

Label 1925.1.w.c
Level $1925$
Weight $1$
Character orbit 1925.w
Analytic conductor $0.961$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -55
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1925,1,Mod(1451,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1925.1451");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1925.w (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.960700149319\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2695.1
Artin image: $S_3\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} - \zeta_{12}^{2} q^{11} - \zeta_{12}^{3} q^{13} + \zeta_{12}^{4} q^{14} - \zeta_{12}^{4} q^{16} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{5} q^{18} + \zeta_{12}^{3} q^{22} + \zeta_{12}^{4} q^{26} + \zeta_{12}^{2} q^{31} + 2 q^{34} - \zeta_{12}^{3} q^{43} - q^{49} + q^{56} - \zeta_{12}^{2} q^{59} - \zeta_{12}^{3} q^{62} - \zeta_{12} q^{63} - q^{64} - q^{71} + \zeta_{12} q^{72} + \zeta_{12}^{5} q^{73} + \zeta_{12}^{5} q^{77} - \zeta_{12}^{2} q^{81} - \zeta_{12}^{3} q^{83} + \zeta_{12}^{4} q^{86} - \zeta_{12}^{5} q^{88} + \zeta_{12}^{4} q^{89} - q^{91} + \zeta_{12} q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{26} + 2 q^{31} + 8 q^{34} - 4 q^{49} + 4 q^{56} - 2 q^{59} - 4 q^{64} - 4 q^{71} - 2 q^{81} - 2 q^{86} - 2 q^{89} - 4 q^{91} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1451.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0 0 0 0 1.00000i 1.00000i 0.500000 0.866025i 0
1451.2 0.866025 + 0.500000i 0 0 0 0 1.00000i 1.00000i 0.500000 0.866025i 0
1726.1 −0.866025 + 0.500000i 0 0 0 0 1.00000i 1.00000i 0.500000 + 0.866025i 0
1726.2 0.866025 0.500000i 0 0 0 0 1.00000i 1.00000i 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
7.c even 3 1 inner
11.b odd 2 1 inner
35.j even 6 1 inner
77.h odd 6 1 inner
385.q odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1925.1.w.c 4
5.b even 2 1 inner 1925.1.w.c 4
5.c odd 4 1 385.1.q.a 2
5.c odd 4 1 385.1.q.b yes 2
7.c even 3 1 inner 1925.1.w.c 4
11.b odd 2 1 inner 1925.1.w.c 4
15.e even 4 1 3465.1.cd.a 2
15.e even 4 1 3465.1.cd.b 2
35.f even 4 1 2695.1.q.a 2
35.f even 4 1 2695.1.q.d 2
35.j even 6 1 inner 1925.1.w.c 4
35.k even 12 1 2695.1.g.a 1
35.k even 12 1 2695.1.g.d 1
35.k even 12 1 2695.1.q.a 2
35.k even 12 1 2695.1.q.d 2
35.l odd 12 1 385.1.q.a 2
35.l odd 12 1 385.1.q.b yes 2
35.l odd 12 1 2695.1.g.b 1
35.l odd 12 1 2695.1.g.e 1
55.d odd 2 1 CM 1925.1.w.c 4
55.e even 4 1 385.1.q.a 2
55.e even 4 1 385.1.q.b yes 2
77.h odd 6 1 inner 1925.1.w.c 4
105.x even 12 1 3465.1.cd.a 2
105.x even 12 1 3465.1.cd.b 2
165.l odd 4 1 3465.1.cd.a 2
165.l odd 4 1 3465.1.cd.b 2
385.l odd 4 1 2695.1.q.a 2
385.l odd 4 1 2695.1.q.d 2
385.q odd 6 1 inner 1925.1.w.c 4
385.bc even 12 1 385.1.q.a 2
385.bc even 12 1 385.1.q.b yes 2
385.bc even 12 1 2695.1.g.b 1
385.bc even 12 1 2695.1.g.e 1
385.bf odd 12 1 2695.1.g.a 1
385.bf odd 12 1 2695.1.g.d 1
385.bf odd 12 1 2695.1.q.a 2
385.bf odd 12 1 2695.1.q.d 2
1155.cg odd 12 1 3465.1.cd.a 2
1155.cg odd 12 1 3465.1.cd.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.1.q.a 2 5.c odd 4 1
385.1.q.a 2 35.l odd 12 1
385.1.q.a 2 55.e even 4 1
385.1.q.a 2 385.bc even 12 1
385.1.q.b yes 2 5.c odd 4 1
385.1.q.b yes 2 35.l odd 12 1
385.1.q.b yes 2 55.e even 4 1
385.1.q.b yes 2 385.bc even 12 1
1925.1.w.c 4 1.a even 1 1 trivial
1925.1.w.c 4 5.b even 2 1 inner
1925.1.w.c 4 7.c even 3 1 inner
1925.1.w.c 4 11.b odd 2 1 inner
1925.1.w.c 4 35.j even 6 1 inner
1925.1.w.c 4 55.d odd 2 1 CM
1925.1.w.c 4 77.h odd 6 1 inner
1925.1.w.c 4 385.q odd 6 1 inner
2695.1.g.a 1 35.k even 12 1
2695.1.g.a 1 385.bf odd 12 1
2695.1.g.b 1 35.l odd 12 1
2695.1.g.b 1 385.bc even 12 1
2695.1.g.d 1 35.k even 12 1
2695.1.g.d 1 385.bf odd 12 1
2695.1.g.e 1 35.l odd 12 1
2695.1.g.e 1 385.bc even 12 1
2695.1.q.a 2 35.f even 4 1
2695.1.q.a 2 35.k even 12 1
2695.1.q.a 2 385.l odd 4 1
2695.1.q.a 2 385.bf odd 12 1
2695.1.q.d 2 35.f even 4 1
2695.1.q.d 2 35.k even 12 1
2695.1.q.d 2 385.l odd 4 1
2695.1.q.d 2 385.bf odd 12 1
3465.1.cd.a 2 15.e even 4 1
3465.1.cd.a 2 105.x even 12 1
3465.1.cd.a 2 165.l odd 4 1
3465.1.cd.a 2 1155.cg odd 12 1
3465.1.cd.b 2 15.e even 4 1
3465.1.cd.b 2 105.x even 12 1
3465.1.cd.b 2 165.l odd 4 1
3465.1.cd.b 2 1155.cg odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1925, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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